» Required Reading » Table of Contents » Chapter 11 11.3 When is a Critical Point a Maximum, Minimum or Saddle? Criterion
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When b is 0, so that the second derivative matrix defined above is diagonal, it is easy to determine when a quadratic has an extremum at its critical point and when a saddle: the extremum happens when ac is positive, so that the determinant, 4ac - b² is positive.
In two dimensions the condition 4ac - b² > 0 is the general condition for an extremum, even when b is not 0. This follows from the fact that by an appropriate rotation of coordinates, which will not affect the determinant, we can make b = 0 happen; after which the condition is that the determinant be positive. Since the determinant hasn’t changed, it must have been positive originally.

In general we can rotate coordinates in any dimension so as to "diagonalize" any symmetric matrix, so that in any dimension we can examine the properties of diagonal quadratics, express them in terms of determinants and use them in general.

For example, in three dimensions to be a minimum at the critical point, the diagonal elements must all be positive, (so that the quadratic has a minimum along any axis)  the two by two submatrices of its matrix of  second derivatives that are obtained by omitting a row and the corresponding column must have positive determinants, and so must the entire matrix.

These statements can be written more succinctly as follows. When the eigenvalues of the second derivative matrix are all non-zero and have the same sign at a critical point, then the critical point is a maximum or a minimum. Otherwise not. (A minimum corresponds to the positive sign as in x² + y² + z². See Section 32.9.

Exercises:

11.1 What are the corresponding conditions for the quadratic to have a maximum at the critical point, in three dimensions?

11.2 Find a symmetric 3 by 3 matrix with positive diagonal elements such that the determinants of the two by two matrices obtained by removing a row and the same column are all positive, but the overall determinant is negative. (hint: make all diagonal elements the same, and all non-diagonal elements the same magnitude (but not sign)). Write down the corresponding quadratic. What is it near its critical point?